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Potential of a dipole at any point

Short answer

An electric dipole is a pair of equal and opposite charges ±q separated by 2a, with dipole moment p = 2qa from −q to +q. For r ≫ a, potential V = p cosθ/(4πε₀ r²). On the axial line, V = ±p/(4πε₀ r²).

Define electric dipole. Determine the electric potential at any point at a distance r from the center of the dipole (also find the potential on the axial line).

Electric dipole

An electric dipole is a pair of equal and opposite point charges +q and −q separated by a small distance 2a. Its dipole moment is p = q(2a), a vector directed from −q to +q.

Potential at a general point

            P
           /|\
        r2/ | \r1
         /  |r  \
       -q---O---+q  
        <-a-> <-a->
        angle θ between OP and p

Let P be at distance r from the centre O, with OP making an angle θ with the dipole axis, and r ≫ a. Let k = 1/4πε₀.

  1. Distance of P from +q: r₁; from −q: r₂.
  2. Potential at P: V = kq (1/r₁ − 1/r₂) = kq (r₂ − r₁) / (r₁ r₂).
  3. For r ≫ a, drop a perpendicular from each charge to line OP: r₁ ≈ r − a cos θ, r₂ ≈ r + a cos θ so r₂ − r₁ ≈ 2a cos θ and r₁r₂ = r² − a² cos² θ ≈ r².
  4. Therefore V = kq (2a cos θ) / r², and with p = 2qa:

V = p cos θ / (4πε₀ r²) = p·r̂ / (4πε₀ r²)

Special cases

  • Axial line (θ = 0° or 180°): V = ± p / (4πε₀ r²). Exactly (without approximation), V = q·2a / [4πε₀ (r² − a²)] on the axis on the +q side.
  • Equatorial line (θ = 90°): V = 0, so the potential is zero everywhere on the perpendicular bisector.

The dipole potential falls as 1/r², unlike the potential of a single charge (1/r).

✓ Answer checked by Claude Opus 5.5 (high effort)

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